General background information about data analysis in disease transmission can e.g. be found in [1]}, [2]}. In the beginning of a pandemic it is important to estimate its transmission characteristics ([3]}) and it is common to replace the unobserved generation time \(G\)  by the serial interval \(S\)  (cf. [4]}, [5]}, [6]}), because this can be observed in clinical studies. However, this practice yields biased estimates (cf. [4]}). The estimation of \(R\)  from observations of the serial interval in particular is discussed in [8]}. As an alternative [5]} suggest the following two step procedure: Firstly, fit a parametric model for the incubation time (cf. [10]}, [3]}) and then, secondly, use this to fit a parametric model to the generation time. [12]} describes how to do non-parametric estimation of the incubation time. Problems related to under-reporting or delays (cf. [13]}) are not considered here because we have data in mind which was collected by researchers rather than observational data from self-reporting. In addition to this type of experimental data, it is also possible to collect larger sets with covariates and use prediction techniques to identify transmission pairs (cf. [14]}). Specific parametric results about the above mentioned quantities for Covid-19 can e.g. be found in [3]}, [5]}, [17]}, [10]}, [19]} and in many other places. The setting described in (REF )-() is close to de-convolution and measurement error settings, classical results about which can be found for example in [20]}, [21]}, [22]}. General background about sieve estimators and series estimation can be found in [23]}, [24]} and the references therein. [25]}, [26]} use sieve estimation based on moment conditions while we start from a likelihood. Further reading about plug-in sieve estimates can be found in [27]}, [28]}.
